Authors: Russell Letkeman
The sequence of sets of Z_n on multiplication where n is a primorial gives us a surprisingly simple and elegant tool to investigate many properties of the prime numbers and their distributions through analysis of their gaps. A natural reason to study multiplication on these boundaries is a construction exists which evolves these sets from one primorial boundary to the next, via the sieve of Eratosthenes, giving us Just In Time prime sieving. To this we add a parallel study of gap sets of various lengths and their evolution all of which together informs what we call the S model. We show by construction there exists for each prime number P a local finite probability distribution and it is surprisingly well behaved. That is we show the vacuum; ie the gaps, has deep structure. We use this framework to prove conjectured distributional properties of the prime numbers by Legendre, Hardy and Littlewood and others. We also demonstrate a novel proof of the Green-Tao theorem. Furthermore we prove the Riemann hypothesis and show the results are perhaps surprising. We go on to use the S model to predict novel structure within the prime gaps which leads to a new Chebyshev type bias we honorifically name the Chebyshev gap bias. We also probe deeper behavior of the distribution of prime numbers via ultra long scale oscillations about the scale of numbers known as Skewes numbers.
Comments: 30 Pages. Submitted to Cambridge University Sigma
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